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Thursday, April 16, 2015

N is for . . .

Okay, break's over. Time to get back to some Mad-Cool-Math Nuggets. Today we're going to delve into the wonderful world of set theory and achieve instant literary success in the process. Ready?

To define a mathematical set, think of all the books in your house. Each book is an element in the collection, or set, known as your library. We'll call this set L. Pick out your favorites and you have a subset of your library we could call F. Pick out the ones you've read more than once and make another subset called R (for repeat).

It's possible F = R, meaning every favorite has been read more than once and every book read more than once is a favorite.

Maybe your particular R is empty, meaning you never read a book more than once. In that case R would be an example of a null set, denoted {}. Another name for this is the empty set.

(If you hate to read and have no books then, like the image above, your L = {}. That, in my opinion, would be a sad null set indeed.)

Null sets, or sets that contain no elements, can be a little weird. They are a subset of every set you can think of, which means they must be a subset of themselves.

Say what? Let's go back to your library, the set L, whether it's empty or not. Consider the books written by my cat as a set called C. I love my cat, but she hasn't written any books. She doesn't even stand on my keyboard to type lsdkghkdjgsl. She's too busy sleeping in my bed all day. So the set C is empty or C = {}.

For C to be a subset of L, every book in C must be in L. Conversely, if C is not a subset of L, then there must be some book in C that is not in L. That's impossible, because C is empty and has no books. So C is a subset of L. (That, my friends, was a proof by contradiction. We mathy folk are tricksy, no?)

Have I blown your mind or bored you to sleep? I do apologize, but I did promise literary success, so let's get to it. Since a null set is a subset of every set, then I can say, without lying, that the set of my published novels is a subset of the books on the New York Times Bestseller List. I can also say that my published novels make a subset of the books that won the Pulitzer. How? Because the set of novels that I have published is a null set. Ta da!

Care to share something about your library set L? Is your subset of digital books bigger or smaller than your subset of paper books?

I admire your ability to understand and explain this logic. I got some of it, kind of. We gave away most all of our books, hard cover and paper back. I have gotten really lazy. I prefer to listen to someone read a book to me. The disadvantage to have an audible version is that I like to see favorite phrases or passages written on the page. For that reason I have actually bought the hard copies of the last three books I have listened to.

I love null sets (and hate when I'm trying to solve matrices), but they can be fun. Among my stunning null sets is a subset of cars built by me that are simultaneously Nascar winners and Formula One champions.

I will do everything in my power to visit commenter's blogs unless I've been abducted by aliens or my children get sick. (If my children get abducted by aliens, I will be very busy, of course, catching up on my sleep.)